Though one can write down such examples, I think the idea of such a list rather misses the point. There are qualitative differences that can be hard to apply to specific examples. We know, thanks to Gompf, that arbitrary finitely presented groups appear as $\pi_1$ of symplectic 4-manifolds of symplectic Kodaira dimension 1 and also of symplectic Kodaira dimension 2.
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Tim PerutzSep 21 '12 at 19:06

We also know that such statements are wildly false for Kaehler surfaces of Kodaira dimension 1 or 2, since e.g. we have only finitely many deformation classes of general type surfaces of fixed $c_1^2$ and $c_2$. Yet it might be hard to decide whether some particular symplectic manifold has a Kaehler structure.
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Tim PerutzSep 21 '12 at 19:06

Mohammad: once you restrict to the simply connected case I can no longer make any such sweeping comments (and I won't attempt to answer in a comment box). I wonder what happens, though, if you embed Gompf's manifolds symplectically into some high-dimensional $\mathbb{C}P^n$ using Gromov-Tischler and then blow up this submanifold.
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Tim PerutzSep 21 '12 at 22:37

could you add some details or references - how are you blowing up? why are the Donaldson hypersurfaces not Kahler?
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Ian AgolSep 22 '12 at 1:24

Agol: in step 1, the symplectic mapping torus $X$ is $S^1$ times the usual one. A general theorem of Gromov-Tischler embeds the resulting (integral) symplectic $2N$-manifold symplectically into $\mathbb{C}P^{2N+1}$. You can blow up along such a submanifold much as you would in Kaehler geometry (see e.g Voisin's book). If $b_1(X)$ is odd, we get a symplectic $2N$-manifold with odd $b_3$. Donaldson hypersurfaces obey the Lefschetz hyperplane theorem, so you can then cut down to 8 dimensions preserving the odd $b_3$.
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Tim PerutzSep 22 '12 at 15:45

It seems interesting to ask what happens if you take e.g. some Noether-violating simply connected symplectic 4-manifold and build a symplectic 8-manifold by this procedure; can it ever be Kaehler?
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Tim PerutzSep 22 '12 at 15:49

There are lots of simply-connected four-dimensional examples in Gompf's symplectic sum paper paper which are shown to be non-Kahler by virtue of violating the Noether inequality (see Theorem 6.2). There are also higher-dimensional examples in the last section of Gompf's paper, including infinitely many simply-connected non-Kahler ones in any even dimension at least 8. Some of these are obtained directly by symplectic sum, and others by taking four-dimensional examples and embedding them in CP^n and blowing up, as suggested by Tim and eigenbunny--in these cases Gompf uses the Hard Lefschetz theorem to prove that the result isn't Kahler.

The standard symplectic surgery operations in four-dimensions (symplectic sum, Luttinger surgery, rational blowdown...) should generally be expected to produce non-Kahler manifolds more often than not, though it's not always feasible to see that the result isn't Kahler--the standard ways of doing so are by the parity of $b_1$ or by the Noether inequality. In particular this paper of Fintushel-Park-Stern gives another large collection of simply connected examples violating the Noether inequality.

One can also show that a symplectic manifold is not Kahler by showing that it is not formal (in the sense of rational homotopy theory). Often nonformal examples also don't satisfy hard Lefschetz (so could instead just be shown to be non-Kahler by that criterion), but there is a nonformal example of Cavalcanti which does satisfy hard Lefschetz.

Wait, Tolman only proves that there are no {\em invariant} Kaehler structure on her examples, not that the examples are not Kaehler. I don't remember Woodward's paper well, but I think it is the same kind of result. The example may admit a Kaehler structure, just not invariant.
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Eugene LermanSep 22 '12 at 12:40